Timeline for How much can one say about this differential equation?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 5, 2014 at 16:59 | comment | added | Robert Bryant | Let $P\to S^1$ be a principal right $G$-bundle with connection $\theta:TP\to\frak{g}$. Pulling $P$ back via the covering map $\pi:\mathbb{R}\to S^1$ yields a (flat) bundle $\tilde P\to\mathbb{R}$ that thus has a global $\tilde\theta$-parallel trivialization $\tilde P\to \mathbb{R}\times G$; i.e., $\tilde\theta$ becomes the canonical left-invariant form on $G$. The deck transformation on $\mathbb{R}\times G$ is $\tau(t,g)=(t{+}2\pi, hg)$ for some $h\in G$ (the holonomy element, unique up to $G$-conjugacy). Conversely, any $h\in G$ defines a bundle with connection over $S^1$ as such a quotient. | |
| Sep 5, 2014 at 16:24 | comment | added | Igor Rivin | Robert, to continue on the density theme, how is the proof trivial? Is it written up somewhere, or is it trivial enough to write in a comment? Otherwise, in fact, in the discrete setting I have rediscovered something similar for graphs invariant under crystallographic groups - I think the continuous version is also known to physicists as "Floquet theory"... | |
| Sep 5, 2014 at 16:03 | comment | added | Robert Bryant | @IgorRivin: I agree that what is basic depends on one's point of view, but the content of Floquet's Theorem really is that connections on $S^1$ are classified up to gauge equivalence by their holonomy. In this latter form, not only is the proof trivial, the generalizations to higher dimensional bases and other groups besides $\mathrm{GL}(n,\mathbb{R})$ and the connections with discrete transformation groups, complex monodromy, period mappings, and a host of other phenomena are made more clear, at least in geometry. | |
| Sep 5, 2014 at 14:57 | comment | added | Igor Rivin | Robert, thanks, though as for nomenclature, I think some would say that "holonomy of connections on $S^1$" is a fancy name for "Floquet theory." | |
| Sep 5, 2014 at 9:44 | comment | added | Robert Bryant | @IgorRivin: Oh, sorry, I guess I should have mentioned that. I just did numerical integration of the matrix ODE, specifically 4th order Runge-Kutta. (Even Euler's method would have sufficed for this level of accuracy, but I had 4RK handy.) By the way, "Floquet Theory" is a fancy name for "holonomy of connections on $S^1$". | |
| Sep 5, 2014 at 0:47 | comment | added | Igor Rivin | Robert, I am being dense, but how do you get the expression for $A(2\pi)?$ | |
| Sep 4, 2014 at 21:50 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed an error in the definition of the fundamental matrix and the consequent numerics
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| Sep 4, 2014 at 21:37 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed some typos I introduced in haste.
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| Sep 4, 2014 at 18:23 | history | answered | Robert Bryant | CC BY-SA 3.0 |